The spectrum of Schottky noise in ion storage rings is analyzed as density fluctuations in effectively one-dimensional plasmas. Strong coupling effects in these plasmas are discussed in relation to experimental observations

Totsuji, Hiroo

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Hiroo Totsuji

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Physical Review A

Spectrum of Schottky noise in ion storage rings

Okayama University Scientific Achievement Repository

Memoirs of the Faculty of Engineering, Okayam a University.VoI.26, No.2, pp.129-138, March 1992Spectrum of Schottky Noise in Ion Storage RingsHiroo TOTSU Jr"(Received February 10 , 1992)SYNOPSISThe spectrum of Schottky noise in ion storage rings is analyzed as densityfluctuations in effectively one-dimensional plasmas. Strong coupling effects inthese plasmas are discussed in relation to experimental observations.1. IntroductionStorage rings of heavy or light IOns are now becoming one of powerful tools forexperiments in the field of atomic physics: When ions are sufficiently cooled downby electron cooling and possibly by laser cooling, longitudinal temperatures of theorder of meV are attained and very precise results can be obtained for various atomicprocesses. These ions with very low temperatures also provide us with a chanceto verify the properties of strongly coupled plasmas which have been predicted bynumerical experiments.Along with ion clouds in ion traps, such as Penning and Paul traps, ions in ion stor-age rings may be regarded as typical examples of the one-component plasma (OCP).The harmonic confining (restoring) force due to alternating quadrupole magneticfields is equivalent to the existence of the uniform background of opposite charges*Department of Electrical and Electronic Engineering129130 Hiroo TOTSUJIaround the center of the beam.The most remarkable phenomenon resulting from strong coupling in these ion beamsmay be crystallization. Crystallization in cold ion systems with finite extension hasbeen first shown by numerical experiments! and then observed as the shell structureof laser cooled ions in the Penning trap.2 These structures have been theoreticallyanalyzed by Barrat and the author and positions and populations of these shells havebeen successfully reproduced by a simple model which takes into account the effectof discreteness or correlation as an essential part of the theory.3,4,5For cold ion beams in ion storage rings, the crystallization has not been confirmed;the transverse temperature is of the order of eV and is much higher than the longi-tudinal one. The maximum value of the coupling parameter r hitherto reported isaround 5, while the crystallization is expected for r f'V 180.6As for dynamic properties, the macroscopic deformation of ion clouds in the Penningtrap has been observed7 and given a theoretical explanation. 8 Microscopic chargedensity fluctuations in ion beams have been observed as the Schottky noise spectrum.It has been reported that the electron cooling or the increase of the number of ionshas strong effects on these spectrum even before crystallization; the appearance of thecollective mode and the reduction of noise power (the integral of frequency spectrum)have been observed.9The purpose of this paper is to analyze the density fluctuation in ion beams and todiscuss the effect of strong coupling on these spectra.2. Ion beam as One-Dimensional PlasmaWe consider an ion beam of radius a circulating in an ion storage ring of radiusR. We regard the beam as straight in the z-direction, while imposing the periodicSchottky Noise in fun Starage Rings 131boundary condition with the period 271"R in this (longitudinal) direction. We denotethe charge of an ion by q and the transverse coordinates by r.When the existence of the grounded vacuum chamber of radius b is taken intoaccount, the coulomb energy of the beam with the number density p(r, z) is writtenasH = ~qJdrdz¢(r, z)p(r, z),where </>( r, z) is the electrostatic potential given byPk,k. =Jdrdzp(r, z)Jo(kr) exp( -ikzz).(1)(2)(3)Here Jo(x) and Io(x) are the Bessel and the modified Bessel functions and kz =271"l/271"R = l/R, l being an integer, is the wave number along the axis: l is called theharmonic number.We now assume that charges in the beam is uniformly distributed in the radialdirection as1p(r, z) = -28(a - r)p(z),71"aJdzp(z) = N,(4)(5)where N is the total number of ions. After integrating over transverse coordinates,we have H in the form(6)with(7)132andHiroo TOTSUjIPh. = Jdzdrp(r, z) exp( -ikzz) = Jdzp(z) exp( -ikzz) (8)Here I1(x), Ko(x), and K1(x) are the modified Bessel functions. We thus have aneffectively one-dimensional plasma with a modified Coulomb interaction V(k z ).When kza ~ 1, the modified interaction reduces to the one-dimensional Coulombinteraction between charged sheets as(9)This one-dimensional Coulomb interaction is cut off at kza f"V 1 by the finiteness ofthe beam radius; when kza ~ 1(10)The effective interaction V(k z ) = V(l), regarded as a function of the harmonic num-ber, is equivalent to the impedance for the beam except for trivial factors. lOIn what follows, we are interested only in z or kz dependence of various quantitiesand denote kz by simply k.The Schottky noise expresses the fluctuation of electrostatic potential near thebeam. Its spectrum is thus directly related to the charge density fluctuation in thebeam which is characterized by static and dynamic form factors.The static form factor S(k) is defined by(11)Here <> denotes the statistical average and the time dependence is made explicitfor Ph. For a random distribution, S(k) = 1. This is also the limiting val ue for largevalues of k.The dynamic form factor S(k, w) is defined bySchottky Noise in Ion Starage Rings1 J .S(k,w) = - dtexp(~wt) < Pk(t)P-k(O) > .21rThese two form factors are related byS(k) = ~ JdwS(k,w).133(12)(13)The Schottky noise spectrum and its integral over the frequency are proportionalto the dynamic and the static form factors, respectively. Note that when the averagevelocity of ions is va, we have to shift the frequency from w to w - kvo = w - lwowhere Wo = voiR.Since our system is classical, the fluctuation-dissipation theorem relates the dynamicform factor to the dielectric response function E( k, w) as(14)where T is the temperature and ~ denotes the imaginary part. Due to (13), the staticform factor is then given bykBT [ 1]S(k) = nV(k) 1- E(k,w = 0) ,where n is the average number density3. Random Phase Approximation(15)(16)The screened response function X.c(k, w) is defined as the response of the density tothe total potential field, the potential due to the sum of external and induced chargedensities. The dielectric response function is related to the screened density responsefunction X.c(k, w) byE(k,w) = 1- V(k)x.c(k,w) (17)134 Hiroo TOTSUJ[In the random phase approximation (RPA) or the mean field approximation, thescreened response function is replaced by the response of the ideal gas XO(k, w) or theVlasov expression as lO ,l1(18)Here m is the mass of an ion and f(p) is the distribution function for momentumnormalized asJdpf(p) = n. (19)As in the case of ordinary plasmas, RPA is expected to give an appropriate descriptionof collective properties of this system in the domain of weak coupling.From (15) and (18), the static form factor is given as1S(k) = 1 + nV(k)jkBT' (20)Here we have used that f(p) is Maxwellian in thermal equilibrium.The reduction of the integrated Schottky noise spectrum results from this expressionfor S(k):10,l1 When the number of ions is sufficiently large or the temperature issufficiently low, nV(k)jkBT becomes comparable with unity and we have decreasedvalues for the static form factor for a given harmonic number (or k), as observed inexperiments.9The solution of the equationE(k,w) = 0 (21)gives the collective mode. For long wavelengths, the dielectric response function (18)is expanded into an asymptotic series as(22)Schottky Noise in Jon Starage RingsThe dispersion relation of the collective mode is thus given bylO,ll1w(k) = ± [n~k)] 2 k.135(23)Since V(k) reduces to a constant for small k, we have a phonon-like dispersion relation.The sound velocity is proportional to the square root of the total number of ions N.This is in agreement with experimental observations.The imaginary part of the frequency of the collective mode, the Landau damping,is proportional to df(w / k) / dp and We have sufficiently small damping whenw/k ~ thermal velocity. (24)(25)The cooling of ions thus leads to sharp peaks for collective modes in the Schottkyspectrum: This behavior has also been observed by experiments.We here note that the condition of the well defined collective mode (24) is also thecondition for the reduction of the total noise power.4. Beyond Random Phase ApproximationLet us now consider the effects of strong coupling which are not, at least in princi-pIe, described by the RPA. Our main concern is the behavior of the Schottky noisespectrum.In this kind of analyses, sum rules sometimes provides us with some useful exactrelationships between various quantities.The compressibility sum rule relates the long wavelength limit of the screened re-sponse function to the compressibility of the system as12l~X.c(k,w= 0) = -n (:;)T'Here P is the pressure. From this relation, we have, instead of the RPA expression(20),136 Hiroo TOTSUJI1S(k) = -l---;(-&P....,...)--n=V(~k)"kBT &n T + kBTWhen our system is weakly coupled and the pressure is given by(26)(27)this result reduces to the RPA value (20). The deviation of the pressure from theideal gas value thus appears in the long wavelength behavior of the noise power.Since the density response function is the Laplace transform of the Poisson bracketbetween spatial Fourier components of density, the high frequency behavior of thedielectric response function is determined by the equal time correlation function oftime derivatives of density fluctuationsYThe simplest of such sum rules may be the f-sum rule which concludes the same formfor f(k, w) as (22) even for strong coupling. The dispersion relation of the collectiveoscillation therefore may not be affected by the strength of coupling in the longwavelength domain. Since the terms proportional to w-4 and higher orders includethe effect of correlation, the dispersion relation in short wavelengths will depend oncorrelation through terms higher order in the wave number.As for the static properties, it is naturally expected that S(k) oscillates as a func-tion of k before approaching unity for k = 00. 14 This comes from liquid or solid likeordering of ions in longitudinal direction: The peak positions are at integral mul-tiples of the inverse of the mean distance between ions. For the harmonic numbercorresponding to these peaks of the static form factor, the integrated noise powerincreases due to strong coupling, in contrast to the reduction in the weak couplingdomain described by RPA.The observation of the enhancement of the noise power, however, seems to bedifficult even for ion beams with sufficiently low temperatures. On the one hand,N f'V 107 or more may be necessary to have large enough noise for observation. OnSchottky Noise in Ion Slara!!e Rin!!s 137the other hand, the maximum of the observable harmonic number may also be limitedand the corresponding wavelength is much larger than the mean ion spacing.When ions are strongly coupled, the frequency of the collective oscillation ap-proaches that of phonons in solids: The range of well defined mode also extendsto short wavelengths comparable with the inverse of mean ionic distance. 14 The ob-servation of this behavior may also be difficult for the same reason described above.5. Concluding RemarksThe nature of the spectrum of the Schottky noise in ion storage rings is analyzed byregarding the beam of cold ions as one-dimensional plasma. The effects of finite beamradius and surrounding grounded vacuum tube are properly taken into account andlong wavelength behavior of the spectrum is derived including the case where ions arestrongly coupled. The analysis of the pressure in the domain of strong coupling is inprogress and its effect on the static form factor will be reported elsewhere.1A. Rahman and J. Schiffer, Phys. Rev. Lett. 57, 1133(1986).2S. 1. Gilbert, J. J. Bollinger, and D. J. Wineland, Phys. Rev. Lett. 60, 2022(1988).3H. Totsuji and J.-L. Barrat, Phys. Rev. Lett. 60, 2484(1988).4H. Totsuji, Phys. Rev. A 38, 5444(1988).!lH. Totsuji, Strongly Coupled Plasma Physics (ed. S. Ichimaru, Elsevier Science138 lliroo TOTSLIJIPublishers, 1990), p.213.6For example, W. L. Slattery, G. D. Doolen, and H. E. DeWitt, Phys. Rev. A26,2255(1982).7D. J. Heinzen, J. J. Bollinger, F. L. Moore, W. M. Hano, and D. J. Wineland,Phys. Rev. Lett. 66, 2080(1991).8D. H. E. Dubin, Phys. Rev. Lett. 66, 2076(1991).9M. Steck, M. Blum, M. Grieser, E. Jaeschke, C. M. Kleffner, D. Kramer, W. Ott,and R. Repnow, Proceedings of the Workshop on Electron Cooling and New CoolingTechniques (World Scientific, 1991), p.64.lOY. Y. Parkhomchuk and D. V. Pestrikov, Sov. Phys. Tech. Phys. 25, 818(1980).11 D. Y. Pestrikov, Proceedings of the Workshop on Electron Cooling and New CoolingTechniques (World Scientific, 1991), p.165.12For example, D. Pines and P. Nozieres, The Theory of Quantum Liquids, Vol. 1(W. A. Benjamin, 1966).13For example, D. Forster, Hydrodynamic Fluctuations, Broken Symmetry, and Cor-relation Functions, (W. A. Benjamin, 1975).14For example, M. Baus and J.-P. Hansen, Physics Report 59, 1(1980).

Spectrum of Schottky Noise in Ion Storage Rings

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Spectrum of Schottky Noise in Ion Storage Rings

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